Abe Mirza Practice Test # 4 Statistic. Hypothesis Testing

Transcription

1 Abe Mirza Pracice Tes # 4 Saisic Topics Hypohesis Tesing Page Problems - 4 Soluions A consuling agency was asked by a large insurance company o invesigae if business majors were beer salespersons ha hose wih oher majors. A sample of 40 salespersons wih a business degree showed ha hey sold an average 0f insurance policies per week wih a s. dev. of.8 policies. Anoher sample of 45 salespersons wih a degree oher han business showed ha hey sold an average of 9 insurance policies per week wih a s. dev. of.35 policies. A.5% significance level, can i be concluded ha persons wih business degrees are beer salespersons han hose who have a degree in anoher area? -Among drivers who have had a car crash in he las year, 88 are randomly seleced and caegorized by age, wih he resuls lised below. If all ages have he same crash rae, we would expeced (because of he age disribuion of licensed drivers) he given caegories o have 6%, 44%, 7% and 3% of he subjecs, respecively. A 0% significance level, es he claim ha he given disribuion of crashes conforms o he disribuion of ages. Age Under Over 64 Toal O(observed)= # of Drivers Many sudens suffer from mah anxiey. A professor who eaches saisics offered her sudens a hour lecure and ways o overcome on mah anxiey. The following able gives he es scores saisics of seven sudens before and afer hey aendan his lecure. Before Afer d=difference Σd = d s d Tes a.5% wheher aending his lecure increases he average score in saisics. 4- A real esae agen claims ha he average price of a -week condominium ime-share on Hilon Head Island, is a mos $9,00. The populaion sandard deviaion is $,00. A sample of 34 such unis had an average selling price of $0,45. Does he evidence suppor he claim a a 0.05? 5- Expers claim ha women commi 0% of murders. Is here enough evidence o rejec his claim if in a sample of 67 murders, women commied 0? Use a significance level of I is claimed ha he average GPA of ARC sudens is a leas.8. A sample of 0 ARC sudens showed an average GPA of.78 wih a sandard deviaion of.6. Use 0.0 o es his claim. 7- I is claimed ha he average GPA of Sierra sudens is more han.8. A sample of 56 Sierra sudens showed an average GPA of.88 wih a sandard deviaion of.5 Use.05 o es his claim. 8- I is claimed ha he average GPA of ARC sudens is.8. A sample of 0 sudens showed he following;.6, 3.,.5,.8,.7, 3.4, 3.,., 3.5,.9. Use of 0.0 o es his claim Pracice Tes 4 /0/0

2 9- The following able shows he number of persons in a random sample of 0 lised according o he day of he week hey prefer o do heir grocery shopping. Day of he Week Mon Tue Wed Thu Fri Sa Sun O(observed) = Shoppers A.5%, es he null hypohesis ha he proporion of persons who prefer o do heir grocery shopping on a paricular day is he same for all days of he week. 0- A real esae agen claims ha he average price of a -week condominium ime-share on Pinole Island, S C is a leas $3,500. The populaion sandard deviaion is $,900. A sample of 4 such unis had an average selling price of $,545. Does he evidence suppor he claim a 0.0? - Expers claim ha women commi 5% of homicides. Is here enough evidence o rejec his claim if in a sample of 05 homicides, women commied 8? Use a significance level of A consuling agency was o asked by a large insurance company o invesigae if business majors were beer salespersons ha hose wih oher majors. A sample of 60 salespersons wih a business degree showed ha hey sold an average 0f 3 insurance policies per week wih a s. dev. of.3 policies. Anoher sample of 75 salespersons wih a degree oher han business showed ha hey sold an average of 6 insurance policies per week wih a s. dev. of.58 policies. A 5%, can i be concluded ha persons wih business degrees are as good salespersons as hose who have a degree in anoher area? 3- There was a research on he weighs a birh of he children of urban and rural women. The researcher suspecs here is a significan difference beween he mean weighs a birh of children of urban and rural women. To es his hypohesis, he selecs independen random samples of weighs a birh of children of mohers from each group, wih he following mean weighs and sandard deviaions. Tes he researcher's belief, a =.0. Urban mohers ( ) n = 75 x 3 s =. Rural mohers ( ) n = 64 x.95 s = Abe Claims ha generally % of his sudens are geing grade of A, 6% B, 4% C, % D and 6 hey % F. The following able liss he grade disribuion for a sample of 00 sudens for sa class Grade A B C D F Toal O(observed)=Sudens A 0% significance level, es Abe s claim smokers were quesioned abou he number of hours hey sleep each day. We wan o es he hypohesis ha he smokers need less sleep han he general public which needs an average of 7.7 hours of sleep. We follow he seps below. If he sample mean is 7.5 and he sandard deviaion is.5, wha can you conclude? 6- Suppose ha you inerview 000 exiing voers abou who hey voed for governor. Of he 000 voers, 450 repored ha hey did no voe for he democraic candidae. Is here sufficien evidence o sugges ha he democraic candidae will win he elecion a he.0 level? 7-The liquid chlorine added o swimming pools o comba algae has a relaively shor shelf live. Records indicae he mean shelf life is 60 hours. Anoher chemical was added o see if i would increase he life. A sample of nine jugs wih he addiive had a sample mean of 7.44 hours wih a sample sandard deviaion of hours. Tes a he 0.05 level wheher he chemical addiive increases he shelf life. 8-Which of he following is no a requiremen for he wo sample es of means for independen samples? a Normal populaions b. Equal populaion sandard deviaions c. Equal sample sizes d. All of he above are required Pracice Tes 4 /0/0

3 9. A bank branch locaed in a commercial disric of a ciy has developed an improved process for servicing cusomers during noon o:00 p.m. peak lunch ime period. The waiing ime of a random sample of 5 cusomers during his hour is recorded over period of a week and he resuls are as follows: 4., 5.55, 3.0, 5.3, 4.77,.34, 3.54, 3.0, 4.50, 6.0, 0.38, 5., 6.46, 6.9, 3.79 a) A 0% significance level, is here evidence ha he mean waiing ime is less han 5 minues? b) Wha assumpions mus hold in order o perform his hypohesis? c) Evaluae his assumpion hrough a graphical approach. d) As a cusomer walks ino he branch office and during he lunch hour, she asks he branch manager how long she can expec o wai? The branch manager replies, Almos cerainly no longer han 5 minues. On he basis of he resul (a), evaluae his saemen. 0. Fory cars from wo differen models of a car manufacurer are compared o see if he smaller model has a higher average MPG han he larger model. From his survey, he sample means and sandard deviaions are 8. and 3.0, and.6 and 5., respecively. Can we conclude ha smaller cars have lower MPG?. An eigh-sided die is rolled 60 imes. The number of s, s, 3 s, 4 s, 5 s, 6 s, 7 s & 8 s obained are 4, 36, 5, 7, 9,, 8 & 9, respecively. We are esing o see if he die is fair. Use.0 for his problem?. Suppose we wan o show ha only children have an average higher choleserol level han he naional average. If he mean choleserol level for all Americans is 90, we es 00 only children and find ha x = 98 and s = 5. Is here evidence o sugges ha only children have an average higher choleserol level han he naional average? 3. Suppose i is claimed ha in a very large bach of componens, abou 0% of iems conain some form of defec. I is proposed o check wheher his proporion has increased, and his will be done by drawing randomly a sample of 50 componens. In he sample, 0 are defecives. Does his evidence indicae ha he rue proporion of defecive componens is significanly larger han 0%? Tes a A deparmen sore believes ha hey ha ge he same number of reurns each day of he week. Based on observed daa and.0, is he deparmen sore correc in heir belief? Reurns M T W R F S S Toal O(observed)=Sudens Currenly, he bes drug for fighing he flu has an average duraion of. days unil he flu sympoms are gone. A new drug claims o be beer. A group of 00 people, who jus acquired he flu, are given his drug. The average duraion of he flu for his group was.05 days, and he sandard deviaion of he duraion for his group was.3 days. Is his drug beer, or is he observed daa due o chance? 6. Currenly, he bes drug for fighing he flu has an average duraion of. days unil he flu sympoms are gone. A new drug claims o be beer. A group of 0 people, who jus acquired he flu, are given his drug. The average duraion of he flu for his group was. days, and he sandard deviaion of he duraion for his group was.3 days. Is his drug beer, or is he observed daa due o chance? 7. According o some sudy, in 003 he average saring salary for hisory major was $6,80 and he saring salary for psychology majors was $5,689. Suppose ha hese mean saring salaries are based on random samples of 05 hisory majors and psychology majors, and furher assume ha he s. deviaion for he saring majors were $3850 and $370, respecively, in 003. Tes a he % significance level wheher he 998 mean saring salary for all hisory majors exceeded ha for all psychology majors. Pracice Tes 4 /0/0 3

4 Soluion : OC: : OC: H 0 : H : H 0 : 0 H : 0 CV Tes Saisics Conclusion Commen Z =.96 Z ( 9) 0 H H 0 : Saed proporions are correc. H : Saed proporions are incorrec. Age Under Over 64 Toal Observed Expeced.6(88).44(88).7(88).3(88) ( O E) (36-4) (-39) (-4) (9-) ( O E) / E 484/4 34/39 44/4 64/ ( O E) = 54.7 E = TS = 54.7 Conclusion: H 0 Commen: H No, age groups have a disproporionae number of crashes. 3 : d 0 OC: d 0 : d 0 H : d (5 0) : 9, 00 OC: 9,00 : 9, 00 H : 9, 00 z.645 z 34( ) : P 0.0 OC: P 0.0 :.8 OC:.8 : P 0.0 H : P 0.0 :.8 H :.8 z Z.335 0(.78.8) (.0) 67 7 :.8 OC:.8 :.8 H :.8 z.96 Z 56(.88.80) :.8 OC::.8 :.8 H : (.89.8) Pracice Tes 4 /0/0 4

5 H 0 : Equal Proporions of shoppers every day of he week. H : Unequal Proporions of shoppers every day of he week. 9 Criical Value from page 5 of he able Day of he week M T W Th F Sa Su Observed Expeced ( O ) ( O E) / E = E (9-30) (7-30) (-30) (6-30) (36-30) (39-30) (4-30) TS = 4.93 so, i exceeds he criical vale TS = 4.93 ( O E) E = Conclusion: H 0 Commen: H 0 : 3, 500 OC: 3,500 : 3,500 H : 3, ( ) : P 0.5 OC: P 0.5 : OC: 3 : OC: : P 0.5 H : P 0.5 H 0 : 0 H : 0 : H : 4 H 0 :Saed proporions are correc. H : Saed proporions are in correc 5 : 7.7 : 7.7 OC: 7.7 H : : P 0.5 OC: P : 60 OC: 60 : P 0.5 H : P 0.5 : 60 H : 60 z.645 z Z (.5) 05 Z (3 6) (3.95) 0 z.58 Z z.645 z ( O E) E ( ) = 0.75 z Z (.5) ( ) c. Equal sample sizes 9- a : 5 OC: 5 : 5 H : (4.87 5) Pracice Tes 4 /0/0 5

6 9- To perform he -es on he populaion mean you mus assume ha he daa are approximaely normally disribued es b 9- Use he same mehod as projec. es for normaliy. c 9- : 5 H 0 : 5 5(4.87 5) d OC: 5 H : : H s l 0 : s l (8..6) z.645 Z 5.98 OC: H s l : s l H 0 : Die is fair. H : Die is no fair Numbers Observed Expeced ( O E) (4-0) (36-0) (5-0) (7-0) (9-0) (-0) (8-0) (9-0) ( O E) / E = TS =.8.07 TS =.8 Conclusion: H 0 Commen: H : 90 OC: 90 H 0 : 90 H : 90 z.645 z 00 (98 90) : P 0. OC: P 0. H 0 : P 0. H : P 0. z.96 Z (.0) H 0 : Same number of reurns each day of he week. H : Differen number of reurns each day of he week Reurns M T W R F S S Observed Expeced E (4-0) (0-0) (-0) (6-0) (6-0) (33-0) (9-0) ( O ) ( O E) / E 36/0 00/0 64/0 6/0 36/0 9/0 8/ = 5.0 ( O E) / E = TS = 5.0 Conclusion: H 0 Commen: H 5 :. OC:. H 0 :. H :. CV= Ts = -5 H 0 H 6. :. H 0 :. I is LTT and he Criical Value a he 5% significance.833 OC:. H :. Tes saisic ha new drug o be beer 0(..) ha falls inside criical region, so we rejec H 0 :. and accep Pracice Tes 4 /0/0 6

7 7. Hisory major ( ) n = 05 x 680 s = 3850 Psychology majors ( ) n = x 5689 s = 370 : H 0 : H 0 : 0 OC: H : H : 0 Noe: in H is more han hen i is a RTT Righ - ailed Tes, Based on =.0 Criical value (From Table) CV=Z =.36 CV=.36 A R ( xx) 0 z s s n n ( ) Falls ouside CR 0.36 Conclusion: or fail o rejec H 0? Ouside CR hen fail o rejec Commen: or rejec? ha he mean saring salary for all hisory majors exceeded ha for all psychology majors. Pracice Tes 4 /0/0 7